Equivalent power of a series and parallel combination of resistors How do you derive the equivalent power for series combination and parallel combination of resistors? I seem to get the same expression for both. This is what I've done:
Series 
Current $I$ is a constant in series combination
$\hspace{1.3cm} P_1=I^2R_1$
$\hspace{1.3cm}P_2=I^2R_2$
Adding the above two equations
$\hspace{1.3cm} P_1+P_2=I^2(R_1+R_2)\\ 
\implies P_{eff}=I^2R_{eff}$
Parallel
Voltage $V$ across the resistors are taken to be a constant
$\hspace{1.3cm} P_1=\frac{V^2}{R_1}$
$\hspace{1.3cm}P_2=\frac{V^2}{R_2}$
Adding the above two equations
$\hspace{1.3cm} P_1+P_2=V^2(\frac{1}{R_1}+\frac{1}{R_2})\\ 
\implies P_{eff}=\frac{V^2}{R_{eff}}$
In both cases I get the same expression for $P_{eff}$. Obviously this is wrong. But I'm not sure where I'm going wrong. Help is appreciated. 
 A: Your derivation is right. why do you think it is wrong? Because you can manipulate to make the second expression same as the first one and since one is in series and the other is parallel, so they must be wrong? But you arrived at the two expressions through  different paths, in the first you took current as constant( series combination) and in the second you took pot. diff. as constant ( parallel combination) . In any combination of resistors, after you find out the equivalent resistance , the total current across the combination and the total potential drop across the combination, you can write the effective power as any of the above two expressions ( given by you). But if you start from individual resistors, you will have to follow diff. paths for different types of combinations, as you have followed. But if you have calculated the equivalent resistance , it does not matter( except while calculating the eq. resistance) how they are combined, the combination can be essentially considered as a single resistor and so the effective power will essentially have the same set of interchangable expression for any single resistor ( originating from a combination of resistors or not). Hope it helps
